By continuity, this result shows that positivity of the kernel \(-k\)  ensures that \(_{\epsilon }\)  is positive for \(\epsilon \)  large enough. It is an open problem to show that this also ensures positivity for all \(\epsilon \) .
Note furthermore that when \(\epsilon =+\infty \) , even if \(_{\infty }\)  is positive, it fails to be definite, i.e. there exists non isometric mm-spaces such that \(_{\infty }(,) = 0\) , see [1]}.
We refer to [2]} for the study of counter-examples and other maps \(\Phi \) .
